Ngô Quốc Anh

May 3, 2011

A squeezing argument and application to logistic equations

Filed under: Uncategorized — Ngô Quốc Anh @ 1:19

The purpose of this note is to introduce the squeezing method. As an application, we prove some Liouville type theorem for solutions to logistic equations. Here the text is adapted from a paper by Du and Ma published in J. London Math. Soc. in 2001 [here].

Let prove the following theorem.

Theorem. Let \lambda \in \mathbb R and u \in C^2(\mathbb R^N) be a non-negative stationary solution of

u_t - \Delta u = \lambda u - u^p.

Then u must be a constant.

The basic ingredients in the proof consist of the following three lemmas. But first of all, let us denote by L a (uniformly) second order elliptic operator satisfying the PDE

-L(u)=\lambda a(x)u-b(x) u^p, \quad x\in \mathbb R^N

where

L(u)=\sum_{ij} (a_{ij}u_{x_i})_{x_j}

with a_{ij} smooth, a_{ij}=a_{ji} and

\sigma_1 |\xi|^2 \leqslant \sum_{ij}a_{ij}(x)\xi_i\xi_j \leqslant \sigma_2 |\xi|^2

for some positive constants \sigma_i, i=1,2.

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